Evaluating Functions

Evaluating Functions

To evaluate a function is to:

Replace (substitute) its variable with a given number or expression.

Like in this example:

Example: evaluate the function f(x) = 2x+4 for x=5

Just replace the variable "x" with "5":

f(5) = 2×5 + 4 = 14

Answer: f(5) = 14

More Examples

Here is a function:

f(x) = 1 - x + x2

Important! The "x" is just a place-holder! And "f" is just a name.

It would be the same function if I wrote:

  • f(q) = 1 - q + q2
  • w(A) = 1 - A + A2
  • h(θ) = 1 - θ + θ2

 

Evaluate For a Given Value:

Let us evaluate that function for x=3:

f(3) = 1 - 3 + 32 = 1 - 3 + 9 = 7

Evaluate For a Given Expression:

Evaluating can also mean replacing with an expression (such as 3m+1 or v2).

Let us evaluate the function for x=1/r:

f(1/r) = 1 - (1/r) + (1/r)2

Or evaluate the function for x=a-4:

f(a-4) = 1 - (a-4) + (a-4)2
  = 1 - a + 4 + a2 - 8a + 16
  = 21 - 9a + a2

Another Example

You can use your ability to evaluate functions to find other answers:

Example: h(x) = 3x2 + ax - 1,

  • You are told that h(3) = 8, can you work out what "a" is?

 

First, evaluate h(3):   h(3) = 3×(3)2 + a×3 - 1
     
Simplify:   h(3) = 27 + 3a - 1
    h(3) = 26 + 3a
     

Now ... we know that h(3) = 8, so:

26 + 3a = 8

Move 26 to other side:   3a = 8 - 26 = -18
     
Divide by 3:   a = -6

 

Check:

h(x) = 3x2 - 6x - 1

h(3) = 3(3)2 - 6·3 - 1 = 27 - 18 - 1 = 8 yes

Careful!

I recommend putting the substituted values inside parentheses () , so you don't make mistakes.

Example: evaluate the function h(x) = x2+2 for x = -3

Replace the variable "x" with "-3":

h(-3) = (-3)2+2 = 9+2 = 11

Without the () you could make a mistake:

h(-3) = -32+2 = -9+2 = -7 (WRONG!)

Also be careful of this:

f(x+a) is not the same as f(x) + f(a)

Example: g(x) = x2

g(w+1) = (w+1)2 = w2 + 2w + 1

g(w) + g(1) = w2 + 12 = w2 + 1

Different Result!